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Strategy 05 of 12Symmetry
If it looks the same from everywhere, use it
WLOGSymmetric FunctionsGeometry
When a problem has symmetry, some variables 'play the same role'. This often simplifies it dramatically. If $x$, $y$, $z$ play the same role, the extremum must be symmetric. Three types: algebraic, geometric, cyclic.
When to use it
- →When variables appear symmetrically
- →When the statement doesn't change under permutation of variables
- →In geometric figures with regular structure
- →When there is periodicity or cyclic structure
How to think (step by step)
- 1Identify the symmetry: Which elements play the same role?
- 2Check: If you swap $x$ and $y$, what changes?
- 3Use the symmetry: The solution must respect it
- 4WLOG: Assume $x \leq y \leq z$ to simplify
Practice Problems
Three problems at increasing difficulty — try each before revealing the hint or solution.
📘Basic
Find the maximum of \(f(x, y, z) = xy + yz + zx\) when \(x, y, z > 0\) with \(x + y + z = 1\).
📙Intermediate
\(n\) people sit in a circle. Each person has a number. Every step, each person replaces their number with the average of their two neighbors' numbers. Prove that after enough steps, all numbers become equal.
📕Advanced
Let \(a, b, c\) be sides of a triangle with \(a+b+c = 1\). Prove: \(a^2+b^2+c^2 < 1/2\).